It is no different from any other part of Moodle, and the answer to your question, as asked, in no.
Conceptually, what happens is this: every time Moodle processes a HTTP request, it connects to the database, does a bunch of things, then disconnects. I say conceptually, because often servers will be set up with connection pooling, so instead of continually connecting and disconnecting, it keeps a few connections around, and keeps re-using them, but that is just a performance optimisation.
